## Fall 2022

### Approximating Eigenvalues via the Landscape function

• Shiwen Zhang, UMass Lowell
• September 14, 12:30-1:30 p.m., In-person and Virtual Meeting

Abstract: In this talk, we study the approximation of eigenvalues via the Landscape function for some random Schrodinger operators as well as some random band matrices. We first give a brief review of the localization landscape theory. Then we focus on some one-dimensional model and show that ratio the ground state energy and the minimum of the landscape potential approaches pi^2/8 as the size of the system approaches infinity. We then present numerical stimulations for the same asymptotic behavior for excited states of some random band matrices. Finally, we discuss conjectures and open problems based on these numerical results and their relation to random graphs as well as higher dimensional models.

### Van der Waerden’s theorem is (nearly) 100: the Ramsey heuristic and the Algebraic Structure of the Stone–Čech Compactification

• John Johnson, Ohio State University
• September 28, 12:30-1:30 p.m., Virtual Meeting

An observation of Kakeya and Morimoto states van der Waerden’s theorem is equivalent to the assertion that “sets of positive integers with bounded gaps (a notion of size) contains arbitrarily long arithmetic progressions.” This observation and related techniques are prototypical of a ubiquitous heuristic: notions of size and their structures imply some interesting (combinatorial) pattern. Motivated by this classical result and its modern generalizations, I’ll illustrate via a suggestive visualization and show how precise instances of this heuristic translate into interesting questions and applications of the algebraic structure of the Stone–Čech compactification. I’ll highlight those generalizations that, so far, seem to require a combination of algebraic and combinatorial techniques.

(Based on previous work with Vitaly Bergelson and Joel Moriera; Cory Christopherson; and Florian Richter.)

### A Simple and Accurate Numerical Method for Singular and Near-singular Integration

• Bobbie Wu, UMass Lowell
• October 12, 12:30-1:30 p.m., In-person and Virtual Meeting

Boundary Value Problems (BVPs) are ubiquitous in engineering and scientific applications. One of the most robust and accurate methods for solving BVPs is the Boundary Integral Equation Method, which has the great advantage of dimensionality reduction: all of the unknowns reside on the boundary surface instead of in the volume. A key challenge when solving integral equations is that special quadrature methods are required to discretize the underlying singular and near-singular integral operators. Accurate discretization of these operators is especially important in, for example, problems that involve close structure-structure or fluid-structure interactions. In this talk, we present some recent advancements on singular and near-singular numerical integration based on the Trapezoidal rule -- one of the simplest quadrature methods.

### A Neural Network Approach for Homogenization of Multiscale Problems

• Jihun Han, Dartmouth College
• October 19, 12:30-1:30 p.m., In-person and Virtual Meeting

Abstract: We propose a neural network-based approach to the homogenization of multiscale problems. The proposed method uses a derivative-free formulation of a training loss, which incorporates Brownian walkers to find the macroscopic description of a multiscale PDE solution. Compared with other network-based approaches for multiscale problems, the proposed method is free from the design of hand-crafted neural network architecture and the cell problem to calculate the homogenization coefficient. The exploration neighborhood of the Brownian walkers affects the overall learning trajectory. We determine the bounds of micro- and macro-time steps that capture the local heterogeneous and global homogeneous solution behaviors, respectively, through a neural network. The bounds imply that the computational cost of the proposed method is independent of the microscale periodic structure for the standard periodic problems. We validate the efficiency and robustness of the proposed method through a suite of linear and nonlinear multiscale problems with periodic and random field coefficients. This is joint work with Yoonsang Lee.

### Self-similar Blowup Phenomena in Nonlinear Evolution Equations

• Irfan Glogic, University of Vienna
• October 26, 12:30-1:30 p.m., Virtual Meeting
Abstract: Nonlinear evolution equations, i.e., nonlinear time-dependent partial differential equations (PDEs) play a central role in the mathematical description of natural phenomena. A distinctive feature of these equations is the possibility of breakdown of solutions in finite time. This phenomenon, which is also called singularity formation or blowup, has both physical and mathematical significance, and as a consequence, predicting blowup and understanding its nature are the central problems of the modern analysis of nonlinear PDEs. In this talk we discuss the question of blowup in the context of two types of evolutions equations, the wave maps equation which comes from physics, and the Keller-Segel system arising in mathematical biology. In both cases, generic blowup phenomena are expected to be driven by the so-called self-similar solutions. In line with this, we outline a general framework for stability analysis of self-similar solutions in these contexts, we discuss the new mathematical problems this approach generates, we mention some results we have obtained recently, and if time permits, we show some numerics which lead to a series of conjectures that will keep us busy in the future.

### Riemann’s Non-differentiable Function: A Turbulent History

• Daniel Eceizabarrena, UMass Amherst
• November 9, 12:30-1:30 p.m., In-person and Virtual Meeting

Abstract: Around 1860, Riemann challenged the beliefs of the time that a continuous function must have a derivative. The function he devised was not given by the typical closed expression, but rather by an infinite sum which he claimed to be continuous everywhere but to have a derivative nowhere. Claim is the correct word, for he did not prove it, leaving with it two long-lasting footprints: a big problem, unsolved until 1970, and a revolution and the consequent establishments of the foundations of modern mathematics. This function came to be known as Riemann's non-differentiable function which, turns out, is almost nowhere differentiable. Recently, this rich analytic structure has proved itself valuable in the setting of turbulence, one of the biggest open problems in mathematical physics. I will begin the talk with a broad historic overview to then describe the structure of the function and the results that were progressively proved for it. After that, I will aim at the role that it plays in turbulence, introducing elements like vortex filaments, multifractality and intermittency, together with the relevant mathematical tools used in their study.

### Improving Numerical Accuracy for the Viscous-Plastic Formulation of Sea Ice

• Tongtong Li, Dartmouth College
• November 16, 12:30-1:30 p.m., In-person and Virtual Meeting

Abstract: Accurate modeling of sea ice dynamics is critical for predicting environmental variables, which in turn is important in applications such as navigating ice breaker ships, and has led to extensive research in both modeling and simulating sea ice dynamics. The most widely accepted model is the one based on the viscous-plastic formulation introduced by Hibler, which is intrinsically difficult to solve numerically due to highly nonlinear features. In particular, sea ice simulations often significantly differ from satellite observations. In this study we focus on improving the numerical accuracy of the viscous-plastic sea ice model. We explore the convergence properties for various numerical solutions of the sea ice model and in particular examine the poor convergence seen in existing numerical methods. To address these issues, we demonstrate that using higher order methods for solving conservation laws, such as the weighted essentially non-oscillatory (WENO) schemes, is critical for numerically solving viscous-plastic formulations whenever the solution is not smooth. Moreover, WENO yields higher order convergence for smooth solutions than standard central differencing does. Our numerical examples verify this, and in particular by using WENO, we are able to resolve the discontinuities in the sharp features of sea ice covers. We also propose an approach utilizing the idea of phase field method to develop a potential function method which naturally incorporates the physical restrictions of ice thickness and ice concentration in transport equations. Our approach results in modified transport equations with extra forcing terms coming from potential energy function, and has the advantage of not requiring any post-processing procedure that might introduce discontinuities and thus ruin the solution behavior.

### Old and New Questions in the Regularity Theory of the d-bar-Neumann Problem

• Gian Maria Dall'Ara, Scuola Normale Superiore Pisa
• November 30, 12:30-1:30 p.m., Virtual Meeting

Abstract: In this talk, I will discuss various questions in the analysis of the d-bar-Neumann problem, a classical noncoercive boundary value problem for the Laplacian whose regularity theory is of great importance in complex analysis and geometry. Despite the deep results established in the last 60 years (starting with seminal work by D. Spencer, J. J. Kohn, J. Nirenberg, E. Stein, C. Fefferman,...), several central questions remain unanswered. I will try to survey classical and more recent approaches to the problem, highlighting the variety of techniques available, ranging from algebraic geometry to mathematical physics.

## Spring 2022

### A Dynamical Approach to Multiplicative Number Theory

• Florian Richter, École Polytechnique Fédérale de Lausanne
• February 2, 2-3 p.m., Virtual Meeting

Abstract: One of the fundamental challenges in number theory is to understand the intricate way in which the additive and multiplicative structures in the integers intertwine. In my talk we will explore a dynamical approach to this topic. More precisely, we will introduce a new framework for treading questions in multiplicative number theory using ideas from ergodic theory. This leads to a new proof of the Prime Number Theorem and to a formulation of an extended form of Sarnak's Mobius randomness conjecture.

### Symmetry in Deep Neural Networks

• Robin Walters, Northeastern University
• February 16, 2-3 p.m., Virtual Meeting

Abstract: Deep neural networks have had transformative impacts in many fields including computer vision, computational biology, and dynamics by allowing us to learn functions directly from data. However, there remain many domains in which learning is difficult due to poor generalization or the need for enormous model sizes. We'll explore two applications of representation theory to neural networks which help address these issues. Firstly, consider the case in which the data represent an $G$-equivariant function. In this case, we can consider spaces of equivariant neural networks which may more easily be fit to the data using gradient descent. Secondly, we can consider symmetries of the function space as well.  Exploiting these symmetries can lead to models with fewer free parameters, faster convergence, and more stable optimization. This is joint work with Rui Wang, Jinxi Li, Rose Yu, and Iordan Ganev.

### Convex relaxations for variational problems arising from pairwise interaction problems

• David Shirokoff, New Jersey Institute of Technology
• March 30, 2-3 p.m., Virtual Meeting

Abstract: We examine the problem of minimizing a class of nonlocal, nonconvex variational problems that arise from modeling a large number of pairwise interacting particles in the presence of thermal noise (i.e., molecular dynamics). Although finding and verifying local minima to these functionals is relatively straightforward, computing and verifying global minima is much more difficult. Global minima (ground states) are important as they characterize the structure of matter in models of self-assembly, as well as (1st order) phase transitions. We discuss how minimizing the functionals can be viewed as testing whether an associated bilinear form is co-positive. We then develop sufficient conditions for global optimality (which in some cases are provably sharp) obtained through a convex relaxation related to the cone of co-positive functionals. The advantage of the convex relaxation is that it results in a conic variational problem that is "computationally tractable", and may be solved using modern numerical techniques. The solutions provide fundamental information on the shapes of self-assembled materials in the corresponding models and phase transitions at zero temperature.

### Knot Invariants, Categorification, and Representation Theory

• Arik Wilbert, University of South Alabama
• April 13, 2-3 p.m., Virtual Meeting

Abstract: This talk provides a survey highlighting connections between representation theory, low-dimensional topology, and algebraic geometry related to my current research. I will begin by recalling basic facts about the representation theory of the Lie algebra sl2 and discuss how these relate to the construction of knot invariants such as the well-known Jones polynomial. I will then introduce certain algebraic varieties called Springer fibers and explain how they can be used to geometrically construct and classify irreducible representations of the symmetric group. These two topics turn out to be intimately related. More precisely, I will demonstrate how one can study the topology of certain Springer fibers using the combinatorics underlying the representation theory of sl2. On the other hand, I will show how Springer fibers can be used to categorify certain representations of sl2. As an application, one can upgrade the Jones polynomial to a homological invariant which distinguishes more knots than the polynomial invariant. At the end of the talk, I will discuss future research directions and explore how this picture might generalize to other Lie types beyond sl2.

### Recent Progress on Mahler’s Problem in Diophantine Approximation

• Osama Khalil, University of Utah
• April 27, 2-3 p.m., Virtual Meeting

Abstract: A classical result of Khintchine’s provides a zero-one law for the Lebesgue measure of points in Euclidean space with a given quality of approximation by rational points. In 1984, Mahler asked whether a similar law holds for Cantor’s middle thirds set. His question is part of a long history of results and conjectures aiming at showing that unlikely intersections between Diophantine sets and natural subsets of Euclidean space only occur for well-understood algebraic reasons. Some of these elementary Diophantine questions ultimately lead to difficult problems at the interface of ergodic theory and spectral theory of automorphic forms. I will describe recent joint work with Manuel Luethi leading to progress towards Mahler’s problem and how it is linked it to a notion of “sparse Hecke operators”.

## Fall 2021

• Tess Anderson, Purdue University
• September 8, 11 a.m. - Noon, Virtual Meeting

Abstract: In analysis and related fields, separating the real numbers into dyadic "chunks" is often used as a key tool in many proofs. Dyadic versions of objects define are often easier to analyze, and in many senses, can completely substitute for the continuous objects. To represent a continuous object as a dyadic one, we need a "distinct dyadic system". In this talk we completely characterize all distinct dyadic systems, motivating their study and underlining some interesting number theory lurking under the surface.

### Byzantine Dispersion on Graphs

• William K. Moses Jr., University of Houston
• September 22, 11 a.m. - Noon, Virtual Meeting

Abstract: In this talk, we consider the problem of Byzantine dispersion and extend previous work along several parameters. The problem of Byzantine dispersion asks: given n robots, up to f of which are Byzantine, initially placed arbitrarily on an n node anonymous graph, design a terminating algorithm to be run by the robots such that they eventually reach a configuration where each node has at most one non-Byzantine robot on it. Previous work solved this problem for rings and tolerated up to n-1 Byzantine robots. In this paper, we investigate the problem on more general graphs. We first develop an algorithm that tolerates up to n-1 Byzantine robots and works for a more general class of graphs. We then develop an algorithm that works for any graph but tolerates a lesser number of Byzantine robots. We subsequently turn our focus to the strength of the Byzantine robots. Previous work considers only "weak" Byzantine robots that cannot fake their IDs. We develop an algorithm that solves the problem when Byzantine robots are not weak and can fake IDs. Finally, we study the situation where the number of the robots is not n but some k. We show that in such a scenario, the number of Byzantine robots that can be tolerated is severely restricted. Specifically, we show that it is impossible to deterministically solve Byzantine dispersion when \lceil k/n \rceil > \lceil (k-f)/n \rceil.

### Fundamentally Semidistributive Lattices

• Emily Barnard, DePaul University
• October 6, 11 a.m. - Noon, Virtual Meeting

Abstract: A partially ordered set is a set together with a relation that allows us to compare some but possibly not all elements. In this talk we discuss a special class of posets called semidistributive lattices. Such lattices are ubiquitous in algebraic combinatorics, in the representation theory of quivers, and the study of polyhedra. Motivated by a classic theorem of Birkhoff which classifies all finite distributive lattice, we will discuss a recent classification of finite semidistributive lattice by Reading, Speyer and Thomas. The talk will be self-contained with many examples and connection to recent research projects. All are welcome.

### de Finetti Lattices and Magog Triangles

• Andrew Beveridge, Macalester College
• October 20, 11 a.m. - Noon, Virtual Meeting

Abstract: The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. (These rules are a special case of de Fineti's axiom from probability.) We give a bijection from a family of poset refinements of $F_{n,2}$ to magog triangles. We then adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles. This talk is based on joint work with Ian Calaway (Stanford University) and Kristin Heysse (Macalester College).

### Geometric Incidence Theory and Uniform Distribution

• Ayla Gafni, University of Mississippi
• November 3, 11 a.m. - Noon, Virtual Meeting

Abstract: The Szemeredi-Trotter Incidence Theorem, a central result in geometric combinatorics, bounds the number of incidences between n points and m lines in the Euclidean plane. Replacing lines with circles leads to the unit distance problem, which asks how many pairs of points in a planar set of n points can be at a unit distance. The unit distance problem breaks down in dimensions 4 and higher due to degenerate configurations that attain the trivial bound. However, nontrivial results are possible under certain structural assumptions about the point set. In this talk, we will give an overview of the history of these and other incidence results. Then we will introduce a quantitative notion of uniform distribution and use that property to obtain nontrivial bounds on unit distances and point-hyperplane incidences in higher-dimensional Euclidean space. This is based on joint work with Alex Iosevich and Emmett Wyman.

### Hilbert Geometries and Entropy Rigidity

• Dave Constantine, Wesleyan University
• December 1, 11 a.m. - Noon, Virtual Meeting

Abstract: Any convex, bounded subset of Euclidean space can be equipped with a metric that reflects the shape of its boundary. This metric (or way of measuring distance) is simple to write down and there are many things you can do with it using only tools from linear algebra. The result is a Hilbert geometry, and these can be both familiar and novel. For instance, the familiar Euclidean and hyperbolic spaces can themselves be seen as specific examples of Hilbert geometries. On the other hand, there is a wide world of Hilbert geometries which are not Euclidean or hyperbolic and share some of the properties of these familiar spaces, but differ from them in others. In this talk I'll start with an introduction to Hilbert geometries accessible to anyone who has had some linear algebra. Then in the second half of the talk I will discuss some recent work (joint with Adeboye and Bray) on entropy rigidity for these geometries. In short, we show that examples which minimize volume entropy (a measurement of the geometric complexity of the geometry) must be very special - in fact they are the familiar hyperbolic space we'll encounter at the start of our discussion.

## Spring 2021

### A Generalized Newton Method with Applications to Lasso

• Phat Vo, Mathematics, Wayne State University
• Tuesday, February 9, 12:30 - 1:30 p.m., Zoom Virtual Meeting

The talk discusses a new algorithm of the Newton-type method based on Mordukhovich second-order subdifferentials for solving a crucial class of Lasso problems, where Lasso stands for Least Absolute Shrinkage and Selection Operator. The Lasso, known also as the l1- regularized least-square optimization problem, was described by Tibshirani and since that has been largely applied to various issues in statistics, machine learning, image processing. The proposed algorithm is computationally implemented via MATLAB, and numerical comparisons between our approach and some different approaches are also provided. This is based on joint work with Pham Duy Khanh, Boris Mordukhovich and Dat Ba Tran.

### Solving a Continuous Multifacility Location Problem by DC Algorithms

• Anuj Bajaj, Mathematics, Wayne State University
• Tuesday, February 16, 12:30 - 1:30 p.m., Zoom Virtual Meeting

We introduce a new approach to solve multifacility location problems, which is based on mixed integer programming and algorithms for minimizing differences of convex (DC) functions. The main challenges for solving the multifacility location problems under consideration come from their intrinsic discrete, nonconvex, and nondifferentiable nature. We provide a reformulation of these problems as those of continuous optimization and then develop a new DC type algorithm for their solutions involving Nesterov's smoothing. The proposed algorithm is computationally implemented via MATLAB numerical tests on both artificial and real data sets. This is based on joint work with B. Mordukhovich, N. M. Nam, and Tuyen Tran.

### Arithmetic Ramsey Theory

• Joel Moreira, Mathematics, The University of Warwick
• Tuesday, February 23, 12:30 - 1:30 p.m., Zoom Virtual Meeting

In this talk I will present a survey of arithmetic Ramsey theory. This fascinating subject, which focuses on patterns which arise in arbitrary finite colourings of the natural numbers, combines ideas and tools from diverse areas of mathematics, such as graph theory, Fourier analysis, geometric group theory, and ergodic theory; and has deep connections with number theory and additive combinatorics. The first half of the talk will be dedicated to the history of this subject, and the second to recent progress on a few simple looking but still open problems.

### Solution to Enflo's Problem

• Paata Ivanishvili, Mathematics, North Carolina State University
• Tuesday, March 2, 12:30 - 1:30 p.m., Zoom Virtual Meeting

Pick any finite number of points in a Hilbert space. If they coincide with vertices of a parallelepiped then the sum of the squares of the lengths of its sides equals the sum of the squares of the lengths of the diagonals (parallelogram law). If the points are in a general position then we can define sides and diagonals by labeling these points via vertices of the discrete cube {0,1}^n. In this case the sum of the squares of diagonals is bounded by the sum of the squares of its sides no matter how you label the points and what n you choose. In a general Banach space we do not have parallelogram law.

Back in 1978 Enflo asked: in an arbitrary Banach space if the sum of the squares of diagonals is bounded by the sum of the squares of its sides for all parallelepipeds (up to a universal constant), does the same estimate hold for any finite number of points (not necessarily vertices of the parallelepiped)? In the joint work with Ramon van Handel and Sasha Volberg we positively resolve Enflo's problem. Banach spaces satisfying the inequality with parallelepipeds are called of type 2 (Rademacher type 2), and Banach spaces satisfying the inequality for all points are called of Enflo type 2. In particular, we show that Rademacher type and Enflo type coincide.

### Quantitative Helly Theorems

• Pablo Soberón, Mathematics, Baruch College, City University of New York
• Tuesday, March 9, 12:30 - 1:30 p.m., Zoom Virtual Meeting

Given a family of convex sets in R^d, how do we know that their intersection has a large volume or a large diameter? A large family of results in combinatorial geometry, called Helly-type theorems, characterize families of convex sets whose intersections are not empty. During this talk we will describe how some bootstrapping arguments allow us to extend classic results to describe when the intersection of a family of convex sets in R^d is quantifiably large.​ The work presented in this talk was done in collaboration with undergraduate students.

### Automatically Generating Machine-Checked Proofs for Validating Programming Languages

• Matteo Cimini, Computer Science, UMass Lowell
• Tuesday, March 16, 12:30 - 1:30 p.m., Zoom Virtual Meeting

'Proofs as programs' is a fascinating connection between mathematical logic and programming languages. This theoretical result provides the foundation for interactive theorem provers (a.k.a. proof assistants). These are sophisticated tools that enable users to write proofs, and check that each step of the proof is justified. In the first part of this talk, I will give an overview of the 'proofs as programs' correspondence and interactive theorem provers. One of the applications of interactive theorem provers is the validation of programming languages: A language designer formalizes a programming language with an interactive theorem prover and then proves specific correctness properties about it. In the second part of this talk, I will briefly describe my ongoing work on automatically generating machine-checked proofs from programming languages definitions.

### How Many Points Do You Need to Guarantee Many Patterns?

• Eyvindur Palsson, Mathematics, Virginia Tech
• Tuesday, March 30, 12:30 - 1:30 p.m., Zoom Virtual Meeting

Finding and understanding patterns in data sets is of significant importance in many applications. One example of a simple pattern is the distance between data points, which can be thought of as a 2-point configuration. Two classic questions, the Erdos distinct distance problem, which asks about the least number of distinct distances determined by N points in the plane, and its continuous analog, the Falconer distance problem, explore that simple pattern. Questions similar to the Erdos distinct distance problem and the Falconer distance problem can also be posed for more complicated patterns such as triangles, which can be viewed as 3-point configurations. In this talk, I will give an introduction to both the discrete and continuous questions, report on recent progress, and share some exciting open questions.

### Playing Games with Lattice Successive Minima

• Tushar Das, Mathematics, University of Wisconsin - La Crosse
• Tuesday, April 13, 12:30 - 1:30 p.m., Zoom Virtual Meeting

We present certain sketches of a program, developed in collaboration with Lior Fishman, David Simmons, and Mariusz Urbanski, which extends the parametric geometry of numbers (initiated by Schmidt and Summerer) to Diophantine approximation for systems of m linear forms in n variables. Our variational principle (arXiv:1901.06602) provides a unified framework to compute Hausdorff and packing dimensions of a variety of sets of number-theoretic interest. All our results have dynamical counterparts via the Dani correspondence principle, e.g. we resolve the Kadyrov-Kleinbock-Lindenstrauss-Margulis conjecture by showing that divergent trajectories of a one-parameter diagonal action on the space of unimodular lattices with exactly two Lyapunov exponents with opposite signs has equal Hausdorff and packing dimensions.

Highlights include the introduction of certain combinatorial objects that we call templates, which arise from a dynamical study of Minkowski’s successive minima in the geometry of numbers; as well as a new variant of Schmidt’s game designed to compute the Hausdorff and packing dimensions of any set in a doubling metric space. The talk will be accessible to students and faculty whose interests contain a convex combination of homogeneous dynamics, Diophantine approximation and fractal geometry. I also hope to present a sampling of open questions and directions that have yet to be explored, some of which may be pursued by either following or adapting the technology described in my talk.

### Musimathical Composing, Playing, and Sounding: Traveling Through, and Hearing Out, Musical Space

• Hubert Ho, Northeastern University
• Tuesday, April 20, 12:30 - 1:30 p.m., Zoom Virtual Meeting

In this talk we will discuss what it means to think musimathically, both as a composer of music and as a music analyst. We will review several models of spatiality and musical distance, identify some applications of these concepts in music. We will also review an original composition that uses a just intonation tuning system, based on ideas about tuning that composer Harry Partch used in his music.

### Analysis and Geometry on Groups: Spectral Gaps and Expander Graphs

• Lam Pham, Mathematics, Brandeis University
• Tuesday, April 27, 12:30 - 1:30 p.m., Zoom Virtual Meeting

Expander families are infinite families of highly connected, yet sparse, finite graphs. They were defined and their existence proved in various contexts by Pinsker (1973) and Komogorov-Barzdin (1967). Beyond their own intrinsic interest, they represented a crucial ingredient to the solution of several open problems, and explicitly constructing expanders is a notoriously difficult problem. Margulis (1973) was the first to explicitly construct expanders, exploiting a connection with the representation theory of semisimple Lie groups and early work of Kazhdan. This led Margulis, Sullivan, and Drinfeld to a solution of the Banach-Ruziewicz problem, and the connection with number theory became clear with a construction of so-called “Ramanujan graphs” subsequently obtained independently by Margulis and Lubotzky-Philips-Sarnak using the Ramanujan-Petersson conjectures.

Expander graphs have a rich history in both pure and applied mathematics, and several open problems remain. This talk will focus on one particular feature of expander graphs, the so-called “spectral gap property” and explain how it can give insight into problems of a more algebraic and geometric nature (dense subgroups in Lie groups, strong approximation in algebraic groups, Diophantine approximation).

## Fall 2020

### Singular Integrals and Patterns in Euclidean Spaces

• Polona Durcik (Mathematics, Chapman University)
• Wednesday, Sept. 23, 3:30-4:30 p.m.
• Location: Virtual

Abstract: We give an overview of some problems in the area of multilinear singular integrals and discuss their connection with questions on point configurations in large subsets of Euclidean space.

### Fourier Series, Modulation Invariance and Multilinear Singular Integrals

• Joris Roos (Mathematics, UMass Lowell)
• Wednesday, Sept. 30, 3:30-4:30 p.m.
• Location: Virtual

Abstract: In this talk I will give an overview of some interrelated problems and results in Euclidean harmonic analysis that I have been interested in. We will begin with Carleson's theorem on pointwise convergence of Fourier series and end with certain multilinear singular integrals involving curvature.

### Wide-Band Butterfly Networks: Sub-Wavelength Imaging via Multi-Frequency Neural Networks

• Leonardo Zepeda-Nunez (Mathematics, University of Wisconsin)
• Wednesday, Oct. 7, 3:30-4:30 p.m.
• Location: Virtual

Abstract: For wave-based inverse problems the resolution is usually limited by the so-called diffraction limit, i.e., the smallest features to be reconstructed cannot be smaller than the smallest wavelength of available data. If one properly restricts the class of features to, for example, point-scatterers, the seminal work of Donoho in the early 90’s demonstrates that the recovery of these sub-wavelength features is tractable. However, algorithms to recover a more general class of structured scatterers containing features below the diffraction limit remains an open question. In this talk we aim to surpass the diffraction limit using deep learning coupled with computational harmonic analysis tools. In particular, I will introduce a new a neural network architecture for inverting wide-band data to recover acoustic scatterers at resolutions finer than the classical limit. The architecture incorporates insights from the butterfly factorization and the Cooley-Tukey algorithm to explicitly account for the physics of wave propagation. The dimensions of the network seamlessly adapt to the desired image resolution, resulting in a number of trainable weights that scale quasi linearly with the image resolution and the data bandwidth. In addition, the data is optimally assimilated across frequencies thus enhancing the stability of the training stage. I will provide the rationale for such construction and showcase its properties for several classes of scatterers with sub-Nyquist features embedded in a known background media. (Joint work with Matthew Li and Laurent Demanet).

### Projection Algorithms for Feasibility and Best Approximation Problems

• Rubén Campoy (Mathematics, University of Girona)
• Wednesday, Oct. 14, 3:30-4:30 p.m.
• Location: Virtual

Abstract: Projection algorithms are powerful tools to solve feasibility problems consisting in finding a point in the intersection of a family of sets. One of the most widely used among them is the so-called Douglas-Rachford (DR) algorithm, due to its successful behavior both in convex and non-convex scenarios. In this talk, we shall introduce the family of projection methods with special attention to DR and some of its applications such as Sudoku puzzles. Then, we shall show how a slight modification of this method yields to a new algorithm with a completely different dynamics: it permits to solve best approximation problems rather than feasibility ones; that is, the algorithm finds, not just a point in the intersection, but the closest one to any given point in the space. This paradigm will be illustrated as an application on the problem of finding probability density functions with prescribed moments. Some other promising numerical results will be also reported.

### The Douglas-Rachford Splitting Algorithm in Optimization

• Hung Phan (Mathematics, UMass Lowell)
• Wednesday, Oct. 21, 3:30-4:30 p.m.
• Location: Virtual

Abstract: We explain several fundamental properties of the Douglas-Rachford algorithm based on monotone operator and fixed point theory. An application in civil engineering design will be briefly presented. A part of the discussed material is suitable for both undergraduate and graduate students.

### It's Hip To Be Square (Part I)

• Annie Raymond (Mathematics, UMass Amherst)
• Wednesday, Oct. 28, 4-5 p.m.
• Location: Virtual

Abstract: Establishing inequalities among graph densities is a central pursuit in extremal graph theory. One way to certify the nonnegativity of a graph density expression is to write it as a sum of squares or as a rational sum of squares. We will be giving a pair of consecutive talks about our ongoing research in this area. In this first part, we will explain how one does so and we will then identify simple conditions under which a graph density expression cannot be a sum of squares or a rational sum of squares.

### It's Hip To Be Square (Part II)

• Amanda Redlich (Mathematics, UMass Lowell)
• Wednesday, Nov. 4, 3:30-4:30 p.m.
• Location: Virtual

Abstract: In this second part, we will talk about an application to Sidorenko's conjecture on the density of bipartite subgraphs.

### Signomial Approximation For A Subclass Of Chance-Constrained Geometric Programs With Exponent Uncertainty

• Belleh Fontem (Manning School of Business, UMass Lowell)
• Wednesday, Nov. 18, 3:30-4:30 p.m.
• Location: Virtual

Abstract: We consider a subclass of maximization geometric programs (GPs) with known coefficient parameters that in each constraint, sum up to less than unity. For such programs, the stochasticity of the real-world may impose practical limitations on the degree to which one can exactly resolve the remaining (i.e., exponent) parameters that help define each constraint's feasible space. Confronted with this challenge, we employ a chance-constrained approach which entails maximizing the GP's objective function subject to a specified minimum probability of being feasible. Then, we derive some theoretical properties of the GP, and propose a decomposition algorithm for finding good feasible solutions.

### Deep Learning is Conquering Human Tasks

• Hong Yu (Computer Science, UMass Lowell)
• Monday, Nov. 23, 3:30-4:30 p.m.
• Location: Virtual

Abstract: How far can it go in medicine? Advances, challenges, and future directions.

### Packings in One, Two, and Three Dimensions: a Macro-Meso-Microscopic View

• Jim Propp (Math, UMass Lowell)
• Friday, Dec. 4, 3:30-4:30 p.m.
• Location: Virtual

Abstract: Hexagonal close-packings are the most efficient way to pack unit disks in R^2. But what do we mean by the "the" in the previous sentence? Are hexagonal close-packings the only optimal packings? If we measure optimality by density, the answer is "no"; in fact, there are far too many density-optimal packings to classify in any meaningful way. This suggests that density is too coarse a notion to capture everything that we mean (or should mean!) by "efficient". To study efficiency, we seek ways to quantify defects in a regular packing. An obstacle here is that common kinds of defects inhabit disparate scales (e.g., point defects are infinitesimal compared to line defects, which in turn are infinitesimal compared to the bulk). This suggests we turn to extensions of the real numbers that include infinitesimal elements (or rather, as turns out to be more helpful, infinite elements). We use a regularization trick to make sense of these ideas (starting in one dimension). This enables us to sharpen our notion of optimal packing so that the optimal disk-packings are provably the hexagonal close-packings and no others. A side-benefit is a natural but apparently new finitely additive, non-Archimedean measure in Euclidean n-space; it agrees with n-dimensional volume when applied to finite regions, but some infinite regions are "more infinite" than others.

### Nearest-neighbor tilings in one and two dimensions

• Ronnie Pavlov (Math, University of Denver)
• Wednesday, Dec. 9, 3:30-4:30 p.m.
• Location: Virtual

Abstract: In one dimension, nearest-neighbor tiling systems are somewhat simple objects; questions about them are often solvable via techniques from linear algebra and/or graph theory, and many of their properties have simple characterizations via definitions from those areas. However, in two (and more) dimensions, nearest-neighbor tiling systems suddenly exhibit incredibly complicated behavior, and their study unavoidably leads to surprising areas such as Turing machines and computability theory. In this talk, I'll describe the contrast between these two worlds and explain how some of these surprising connections arise.

## Spring 2020

### Polynomial time guarantees for the Burer-Monteiro method

• Diego Cifuentes, Mathematics, MIT
• Monday, February 24, 3-4 p.m., Location: Olney 430

The Burer-Monteiro method is one of the most widely used techniques for solving large-scale semidefinite programs (SDP). The basic idea is to solve a nonconvex program in Y, where Y is an n×p matrix such that X = YYT. We show that this method can solve SDPs in polynomial time in an smoothed analysis setting. More precisely, we consider an SDP whose domain satisfies some compactness and smoothness assumptions, and slightly perturb the cost matrix and the constraints. We show that if $p\geq\sqrt{(2+2\eta)m}}$, where $m$ is the number of constraints and $\eta>0$ is any fixed constant, then the Burer-Monteiro method can solve SDPs to any desired accuracy in polynomial time, in the setting of smooth analysis. Our bound on $p$ approaches the celebrated Barvinok-Pataki bound in the limit as $\eta$ goes to zero, beneath which it is known that the nonconvex program can be suboptimal.

### Spectral graph theory in quantum communication

• Gabor Lippner, Northeastern University
• Monday, March 2, 4-5 p.m., Location: Olney 430

Physically transmitting quantum information is an important building block of any quantum computer. A possible method to accomplish this is via a "quantum wire", that is, a network of interconnected (coupled) quantum particles. Finding network structures that propagate quantum information efficiently turns out to be very challenging. In this talk I will explain the relevance of spectral graph theory to this problem, and outline some recent results as well as some open problems. Joint work with Mark Kempton, and in part with Shing-Tung Yau, Or Eisenberg, and Whitney Drazen.

### The following seminars have been canceled:

• Mark Lyon, Department of Mathematics and Statistics, University of New Hampshire
• Monday, March 23, 4-5 p.m., Location: Olney 430
• Lam Pham, Brandeis University
• Monday, March 30, 4-5 p.m., Location: Olney 430
• Rubén Campoy, Mathematics, UMass Lowell
• Monday, April 6, 4-5 p.m., Location: Olney 430
• Belleh Fontem, Manning School of Business, UMass Lowell
• Monday, April 13, 4-5 p.m., Location: Olney 430
• Victor Churchill, Dartmouth College
• Monday, April 27, 4-5 p.m., Location: Olney 430

## Fall 2019

### Stochastic Superparameterization Through Local Data Generation

• Yoonsang Lee, Department of Mathematics, Dartmouth College
• Wednesday, September 11, 3:30 p.m., Location: Olney 430

Stochastic superparameterization is a class of multiscale methods that approximate large-scale dynamics of complex dynamical systems such as turbulent flows. Unresolved sub-grid scales are modeled by a cheap but robust stochastic system that mimics the true dynamics of the sub-grid scales, which is crucial to model non-trivial and non-equilibrium dynamics. In this talk, we propose a numerical procedure to estimate the modeling parameters, which avoids the use of climatological data.

### Hardness Results for Sampling Connected Graph Partitions with Applications to Redistricting

• Daryl Deford, MIT
• Wednesday, September 18, 4 p.m., Location: Olney 430

The problem of constructing ”fair'' political districts and the related problem of detecting intentional gerrymandering has received a significant amount of attention in recent years. A key problem in this area is determining the expected properties of a representative districting plan as a function of the input geographic and demographic data. A natural approach is to generate a comparison ensemble of plans using MCMC and I will present successful applications of this approach in both court cases and legislative reform efforts. However, our recent work has demonstrated that the commonly used boundary-node flip proposal can mix poorly on real-world examples. In this talk, I will present some new proposal distributions for this setting and discuss some related open problems concerning mixing times and spanning trees. I will also discuss some generic hardness results for sampling problems on partitions of planar graphs.

### Multiscale Convergence Properties for Spectral Approximations of a Model Kinetic Equation

• Zheng Chen, Department of Mathematics, UMass Dartmouth
• Thursday, October 3, 2 p.m., Location: Olney 430

We prove some convergence properties for a semi-discrete, moment-based approximation of a model kinetic equation in one dimension. This approximation is equivalent to a standard spectral method in the velocity variable of the kinetic distribution and, as such, is accompanied by standard algebraic estimates of the form $N^{-q}$, where $N$ is the number of modes and $q$ depends on the regularity of the solution. However, in the multiscale setting, we show that the error estimate can be expressed in terms of the scaling parameter $\epsilon$, which measures the ratio of the mean-free-path to the domain in the system. In particular we show that the error in the spectral approximation is $\mathcal{O}(\epsilon^{N+1})$. More surprisingly, the coefficients of the expansion satisfy some super convergence properties. In particular, the error of the $\ell^{th}$ coefficient of the expansion scales like $\mathcal{O}(\epsilon^{2N})$ when $\ell =0$ and $\mathcal{O}(\epsilon^{2N+2-\ell})$ for all $1\leq \ell \leq N$. This result is significant, because the low-order coefficients correspond to physically relevant quantities of the underlying system. Numerical tests will also be presented to support the theoretical results.

### Jump Process Approximation of Particle-Based Stochastic Reaction-Diffusion Models

• Samuel Isaacson, Department of Mathematics and Statistics, Boston University
• Wednesday, November 6, 4 p.m., Location: Olney 430

High resolution images of cells demonstrate the highly heterogeneous nature of the nuclear and cytosolic spaces. We are interested in understanding how this complex environment influences the dynamics of cellular processes. To investigate this question we have developed the convergent reaction-diffusion master equation (CRDME), a lattice particle-based stochastic reaction-diffusion method that can model the spatial transport and reactions of molecules within domains derived from imaging data. In this talk I will introduce the CRDME, and explain how it is similar in spirit to the popular reaction-diffusion master equation (RDME) model. The CRDME allows for the reuse of the many extensions of the RDME developed to facilitate modeling within biologically realistic domains, while eliminating one of the major challenges in using the RDME model.

### Geometric Structure in Dependence Models and Applications

• Elisa Perrone, Department of Mathematical Sciences, UMass Lowell
• Wednesday, November 20, 4 p.m., Location: Olney 430

The growing availability of data makes it challenging yet crucial to model complex dependence traits. For example, hydrological and financial data typically display tail dependences, non- exchangeability, or stochastic monotonicity. Copulas serve as tools for capturing these complex traits and constructing accurate dependence models which resemble the underlying distributions of data. This talk inquires into the geometric properties of dependence models and copulas to address statistical challenges in several applications, such as hydrology and weather forecasting. In particular, we study the class of discrete copulas, i.e., restrictions of copulas to uniform grid domains, which admits representations as convex polytopes. In the first part of this talk, we give a geometric characterization of discrete copulas with desirable stochastic constraints in terms of the properties of their associated convex polytopes. In doing so, we draw connections to the popular Birkhoff polytopes, thereby unifying and extending results from both the statistics and the discrete geometry literature. We further consolidate the statistics/discrete geometry bridge by showing the significance of our geometric findings to construct entropy-copula models useful in hydrology. In the second part of this talk, we focus on weather forecasting problems. We show that discrete copulas are powerful tools for empirical modeling in applications. We discuss their use in the context of statistical postprocessing of ensemble weather forecasts, and present a case study for temperature forecasts in Austria.

### Dimension of Sumsets of Restricted Digit Cantor Sets in the Integers

• Daniel Glasscock, Department of Mathematical Sciences, UMass Lowell
• Wednesday, December 4, 4 p.m., Location: Olney 430

Harry Furstenberg made a number of conjectures in the 60's and 70’s seeking to make precise the heuristic that there is no common structure between digit expansions of real numbers in different bases. Recent solutions to conjectures of his concerning the dimension of sumsets and intersections of p- and q- invariant sets now shed new light on old problems. In this talk, I will explain how to use tools from fractal geometry and uniform distribution to determine the dimension of sumsets of restricted digit Cantor sets with respect to different bases in the integers. This talk is based on joint work with Joel Moreira and Florian Richter.

## Spring 2019

#### Computational Plasma Physics in the Solar System and Beyond

• Ofer Cohen, Department of Physics, UMass Lowell
• February 13, 3-4 p.m. Room: Olney 430

#### Applications of Statistical Modeling in Cognitive Neuroscience

• Kensuke Arai, Department of Mathematics and Statistics, Boston University
• February 19, 3:30-4:30 p.m. Room: Olney 430

#### Undergraduate Special Seminar: Cost Estimator Career

• Ryan Porter, Hanscom Air Force Base
• March 19, 2-2:30 p.m. Room: Olney 430

#### The Barycenter Method for Direct Optimization: an Overview, with Applications to Estimation of Switched Linear Models

• Felipe Pait, Electrical Engineering, Universidade de Sao Paulo
• March 25, 4-5 p.m. Room: Olney 430

#### Hardy Spaces of Fuchsian Groups

• Alexander Kheifets, Mathematical Sciences, UMass Lowell
• April 17, 4-5 p.m. Room: Olney 430

#### Evaluation of Far-field Gravitational-Wave Signals From Near-field Data

• Scott Field,Mathematics, UMass Dartmouth
• April 24, 4-5 p.m. Room: Olney 430

#### Master Thesis Defense: Accurate Numerical Solutions of Helmholtz Equation using Layered Media Green's Function

• Djeneba Kassambara, Mathematical Sciences, UMass Lowell
• April 25, 2-3 p.m. Room: Olney 430